LMFDB - L-function 2-1183-13.12-c1-0-13

Dirichlet series

L(s) = 1

− 2.34i·2^-s − 1.14·3^-s − 3.48·4^-s + 1.34i·5^-s + 2.68i·6^-s − i·7^-s + 3.48i·8^-s − 1.68·9^-s + 3.14·10^-s + 1.14i·11^-s + 4.00·12^-s − 2.34·14^-s − 1.53i·15^-s + 1.19·16^-s − 5.83·17^-s + 3.94i·18^-s + ⋯

L(s) = 1

− 1.65i·2^-s − 0.661·3^-s − 1.74·4^-s + 0.600i·5^-s + 1.09i·6^-s − 0.377i·7^-s + 1.23i·8^-s − 0.561·9^-s + 0.994·10^-s + 0.345i·11^-s + 1.15·12^-s − 0.626·14^-s − 0.397i·15^-s + 0.299·16^-s − 1.41·17^-s + 0.930i·18^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1183$ = $7 \cdot 13^{2}$
Sign:	$0.554 + 0.832i$
Analytic conductor:	$9.44630$
Root analytic conductor:	$3.07348$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1183} (337, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1183,\ (\ :1/2),\ 0.554 + 0.832i)$

Particular Values

$L(1)$	$\approx$	$0.8540017559$
$L(\frac12)$	$\approx$	$0.8540017559$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	7	$ 1 + iT $
	13	$ 1 $
good	2	$ 1 + 2.34iT - 2T^{2} $
	3	$ 1 + 1.14T + 3T^{2} $
	5	$ 1 - 1.34iT - 5T^{2} $
	11	$ 1 - 1.14iT - 11T^{2} $
	17	$ 1 + 5.83T + 17T^{2} $
	19	$ 1 - 3.34iT - 19T^{2} $
	23	$ 1 - 3.17T + 23T^{2} $
	29	$ 1 - 10.4T + 29T^{2} $
	31	$ 1 + 1.63iT - 31T^{2} $
	37	$ 1 - 8.51iT - 37T^{2} $
	41	$ 1 - 0.292iT - 41T^{2} $
	43	$ 1 - 8.15T + 43T^{2} $
	47	$ 1 + 10.6iT - 47T^{2} $
	53	$ 1 + 0.782T + 53T^{2} $
	59	$ 1 - 12.6iT - 59T^{2} $
	61	$ 1 + 2T + 61T^{2} $
	67	$ 1 - 6.10iT - 67T^{2} $
	71	$ 1 + 1.53iT - 71T^{2} $
	73	$ 1 + 15.3iT - 73T^{2} $
	79	$ 1 - 0.882T + 79T^{2} $
	83	$ 1 - 12.1iT - 83T^{2} $
	89	$ 1 - 5.73iT - 89T^{2} $
	97	$ 1 - 5.34iT - 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−10.13265296431857231786795846066, −9.064477325019089716676550217134, −8.365808810663076240286476771318, −6.97708655838298592690091816441, −6.30297192413258975543532185306, −4.98676966003366157931403011100, −4.28484594528237478590613509647, −3.13241783977242895236028879296, −2.37834584677918873252825072152, −0.948929377273701275357502308683, 0.53283873545571116694495164634, 2.72385116470159023282838305025, 4.51814284840936683721311782164, 4.95055087164494135194294503703, 5.85319681112244792309232011077, 6.46781861070774953623303540192, 7.18627088516210770829329408257, 8.361183454882534566052187893446, 8.766010580518245499319259348948, 9.368872309283232635005036732334

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8540017559\)
\(L(\frac12)\)	\(\approx\)	\(0.8540017559\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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